The first step in syncing images is to make sure you're signed in to the same Adobe account on both your mobile device running capture, and your desktop running Illustrator.
The second step is to make sure you are looking at the same Library both in Capture and in Illustrator. The default is "My Library" but you can create your own.
The third step is to recognize what can be synchronized with Illustrator:
- Colors - will show up in Illustrator
- Shapes - will show up as Graphics in Illustrator
- Brushes - will show up but are only available to use if you create the brush as an Illustrator type of brush - otherwise they are greyed out
- Looks - will show up in Illustrator but are greyed out an not available to use
Finally, the fourth step is to look in the Libraries panel in Illustrator (Window > Libraries). Make sure the library is set to the one you want (in my example below, I'm showing my "Around Town" library, Here you can see the color themes, a brush, a shape and my looks (which are not available for AI).
I hope this helps!